Proposition 2

The relation « there exists a quadratic form Q over A, non-degenerate and of finite dimension, such that X = θ(Q) » is collectivizing in X (Ens., chap. II, § 1, n° 4).

Indeed, let V be an infinite-dimensional vector space over A, $\mathfrak{S}$ the set of non-degenerate quadratic forms defined on the finite-dimensional subspaces of V, and $\mathfrak{W}$ the set of the $\theta(Q)$ for $Q \in \mathfrak{S}$. It is clear that every non-degenerate quadratic form $Q'$ of finite dimension over A is equivalent to at least one element of $\mathfrak{S}$; whence $\theta(Q') \in \mathfrak{W}$, which proves our assertion.